In digital communication systems, data is encoded into signals using a digital modulation technique before being transmitted using a communication channel. A digital modulation scheme is typically defined by a signal constellation in a complex plane or in a higher dimensional space, wherein each signal point of the constellation corresponds to a data symbol. Once a receiver demodulates the received signal, the transmitted constellation point may be retrieved using the received signal, based on the signal constellation's geometry.
In view of the growing demand for spectral efficiency in digital communications, high order modulation design has received a considerable attention in the recent past. To cater to this demand, modulation and coding, MODCOD, configurations employing constellations of orders 128 and 256 were adopted in the new digital video broadcasting standard (DVBS2X), see ETSI 302-307-2, “Digital Video Broadcasting (DVB) Part II: DVBS2-Extensions (DVB-S2X))”, 2014. Such high order constellations are also under considerations by consultative committee for space data systems, CCSDS, for future standards. Given the fact that the traffic demand for satellite broadband is expected to grow six-fold by 2020, and the continuous need for higher data rates in satellite communications, even larger constellations may be needed in near future.
A distinctive property of satellite communication systems, is the non-linear characteristics of high power amplifier, HPA, on-board of the satellite. Hence, the conventional additive white Gaussian noise, AWGN, channel with average power limitation is no longer an accurate model and nonlinear characteristics have to be taken into account. This usually leads to assume a peak power limited signaling due to the HPA operating near the saturation limits. Under the peak power constraint, the capacity achieving input signal distribution is known to be discrete in amplitude (having finite number of mass points) with a uniformly distributed phase. Even though the optimal distribution for a finite set is not known in general, several previous studies indicate that amplitude and phase-shift keying (APSK) modulations perform very close to the capacity, see for example Kayhan et al, “Joint signal-labeling optimization under peak power constraint,” Int. J. of Satellite Communications and Network, DOI: 10.1002/sat.1016, 2012. Recently, it has been shown that asymptotically, APSK constellations can also achieve the Gaussian capacity, see H. Meric, “Approaching the Gaussian Channel Capacity With APSK Constellations,” in IEEE Communications Letters, vol. 19, no. 7, pp. 1125-1128, July 2015.
Optimizing the APSK signal constellation has been studied by several authors in the literature. The number of points on each circle, the radii and phases of each concentric circle of APSK constellations need to be optimized in order to achieve near capacity performances. Even though for constellations with up to 64 points, the number of possibilities is rather limited and therefore the optimization problem can be still handled, for larger constellations the problem becomes complex and the proposed algorithms in the literature usually fail to provide a good sub-optimal solution.
Another problem which arises regarding the optimization of APSK constellation is the bit to symbol mapping. This problem becomes particularly important in the pragmatic systems, where no iteration between the detector and the decoder is allowed.
Following this observation, several algorithms have been proposed in the literature for constellation design without imposing any particular structure. Recently, a simulated annealing algorithm has been implemented to maximize the average mutual information, AMI, or the pragmatic average mutual information, PAMI, (also known as the bit-interleaved coded modulation (BICM) capacity). When AMI and PAMI can be computed with a good approximation, they provide a reliable metric to assess the performance of the optimized constellation set. When the objective function is PAMI, the joint signal and labeling optimization is needed to secure a good result.
Simulated annealing, see for example Kirkpatrick et al., “Optimization by simulated annealing,” Science, vol. 220, pp. 671-680, May 1983, becomes inefficient for constellations larger than 64. For larger constellations, however, the simulated annealing becomes extremely slow and may converge to a local maximum which is only slightly better than the initializing constellation.
Anedda et. al disclose an APSK signal constellation for modulating data into signal constellation symbols in “64-APSK Constellation and Mapping Optimization for Satellite Broadcasting Using Genetic Algorithms”, IEEE Transactions on Broadcasting, vol. 62, no. 1, 1 Mar. 2016, pages 1-9.